본문 바로가기
KERD
논문KCI 등재

Existence and multiplicity of solutions for nonlinear Schr\"{o}dinger-Kirchhoff-type equations

Haibo Chen|Hongliang Liu|Liping Xu
대한수학회 (The Korean Mathematical Society)|2016.01|대한수학회지 (Journal of the KMS)|제53권 제1호|pp. 201-215

1

조회

0

다운로드

0

인용

0

참고문헌

키워드

genusMorse theoryvariational methodsSchr\"{o}dinger-Kirchhoff-typecritical groups
주제분류수학

초록

In this paper, we consider the following Schr\"{o}dinger-Kirch\-hoff-type equations \[ \left[a+b\left(\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+V(x)|u|^{2})dx\right)\right][-\Delta u+V(x)u]=f(x,u), ~~\mbox{in}~ \mathbb{R}^{N}. \] Under certain assumptions on $V$ and $f$, some new criteria on the existence and multiplicity of nontrivial solutions are established by the Morse theory with local linking and the genus properties in critical point theory. Some results from the previously literature are significantly extended and complemented.

Abstract

In this paper, we consider the following Schr\"{o}dinger-Kirch\-hoff-type equations \[ \left[a+b\left(\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+V(x)|u|^{2})dx\right)\right][-\Delta u+V(x)u]=f(x,u), ~~\mbox{in}~ \mathbb{R}^{N}. \] Under certain assumptions on $V$ and $f$, some new criteria on the existence and multiplicity of nontrivial solutions are established by the Morse theory with local linking and the genus properties in critical point theory. Some results from the previously literature are significantly extended and complemented.

참고문헌 (25)

  1. C. O. Alves, "Positive solutions for a quasilinear elliptic equation of Kirchhoff type", Comput. Math. Appl, 49(1), pp.85~93 (2005)
  2. T. Bartsch, "Critical point theory for asymptotically quadratic functionals and applications to problems with resonance", Nonlinear Anal, 28(3), pp.419~441 (1997)
  3. V. Benci, "An eigenvalue problem for the Schr¨odinger-Maxwell equations", Topol. Methods Nonlinear Anal, 11(2), pp.283~293 (1998)
  4. K. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems" (1993), Birkh¨auser
  5. P. Chen, "Infinitely many solutions for Schr¨oginger-Maxwell equations with indefinite sign subquadratic potentials", Appl. Math. Comput, 226, pp.492~502 (2014)
  6. B. Cheng, "Existence results of positive solutions of Kirchhoff type problems", Nonlinear Anal, 71(8), pp.4883~4892 (2009)
  7. L. Duan, "Infinitely many solutions for sublinear Schr¨oginger-Kirchhoff-type equations with general potentials", Results Math, 66(1-2), pp.181~197 (2014)
  8. X. He, "Infinitely many positive solutions for Kirchhoff-type problems", Nonlinear Anal, 70(3), pp.1407~1414 (2009)
  9. X. He, "Existence and concentration behavior of positive solutions for a Kirchhoff equations in R3", J. Differential Equations, 252(2), pp.1813~1834 (2012)
  10. G. Kirchhoff, "Mechanik" (1883), Teubner
  11. G. Li, "Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3", J. Differential Equations, 257(2), pp.566~600 (2014)
  12. Y. Li, "Existence of a positive solution to Kirchhoff type problems without compactness conditions", J. Differential Equations, 253(7), pp.2285~2294 (2012)
  13. J. L. Lions, "Contemporary Developments in Continum Mechanics and Partial Differential Equations", pp.247~262 (1992)
  14. W. Liu, "Multiplicity of high energy solutions for superlinear Kirchhoff equations", J. Appl. Math. Comput, 39(1-2), pp.473~487 (2012)
  15. Z. Liu, "Existence and multiplicity of solutions for a class of sublinear Schr¨odinger-Maxwell equations", Taiwanese J. Math, 17(3), pp.857~872 (2013)
  16. A. Mao, "Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition", Nonlinear Anal, 70(3), pp.1275~1287 (2009)
  17. J. Mawhin, "Critical point theory and Hamiltonian systems" (1989), Springer
  18. K. Perera, "Nontrivial solutions of Kirchhoff-type problems via the Yang index", J. Differential Equations, 221(1), pp.246~255 (2006)
  19. P. Rabinowitz, "Minimax methods in critical point theory with applications to differential equations", CBMS Reg. Conf. Ser. in Math, 65 (1986), Amer. Math. Soc
  20. A. Salvatore, "Homoclinic orbits for a special class of nonautonomous Hamiltonian systems", Proceedings of the Second World Congress of Nonlinear Analysis, Part 8(Athens, 1996), (8), pp.4849~4857 (1997)
  21. J. Su, "An elliptic resonance problem with multiple solutions", J. Math. Anal. Appl, 319(2), pp.604~616 (2006)
  22. J. Sun, "Ground state solutions for an indefinite Kirchhoff type problem with steep potential well", J. Differential Equations, 256(4), pp.1771~1792 (2014)
  23. X. Wu, "Existence of nontrivial solutions and high energy solutions for Schr¨odingerKirchhoff-type equations in RN", Nonlinear Anal. Real World Appl, 12(2), pp.1278~1287 (2011)
  24. X. Wu, "High energy solutions of systems of Kirchhoff-type equations in RN", J. Math. Phys, 53(6), pp.1~18 (2012)
  25. J. Zhang, "Multiple nontrivial solutions for some fourth-order semilinear elliptic problems", Nonlinear Anal, 60(2), pp.221~230 (2005)

관련 논문AI추천

Existence and multiplicity results for a new Kirchhoff type problem with critical exponent and indefinite potentialXiaotao Qian, Zhigao Shi대한수학회 · 대한수학회보2025 · 제62권 제2호Existence and multiplicity of solutions for Kirchhoff-Schr\"{o}dinger-Poisson system with concave and convex nonlinearitiesGuofeng Che, Haibo Chen대한수학회 · 대한수학회지2020 · 제57권 제6호Infinitely many solutions for a class of modified nonlinear fourth-order elliptic equations on RnGuofeng Che, Haibo Chen대한수학회 · 대한수학회보2017 · 제54권 제3호Multiple solutions for a class of quasilinear Schr\"{o}dinger system in $\mathbb{R}^N$Caisheng Chen, Qiang Chen대한수학회 · 대한수학회보2016 · 제53권 제6호Ground state sign-changing solutions for a class of Schr\"{o}dinger-Poisson-Kirchhoff type problems with a critical nonlinearity in $\mathbb{R}^{3}$Aixia Qian, Mingming Zhang대한수학회 · 대한수학회지2021 · 제58권 제5호Positive solution and ground state solution for a Kirchhoff type equation with critical growthAixia Qian, Caixia Chen대한수학회 · 대한수학회보2022 · 제59권 제4호

이 간행물의 다른 논문 (14건)

$\mathcal{W}$-resolutions and Gorenstein categories with respect to a semidualizing bimodulesXiaoyan Yangpp. 1-17Second-order symmetric duality in multiobjective programming over conesTilak Raj Gulati, Geeta Mehndirattapp. 19-25Global existence and blow-up for a degenerate reaction-diffusion system with nonlinear localized sources and nonlocal boundary conditionsFei Liangpp. 27-43On the strong law of large numbers for weighted sums of negatively superadditive dependent random variablesAiting Shenpp. 45-55Maximal inequalities and an application under a weak dependence신동완, 황은주pp. 57-72
원문 보기